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% Created on 2008-03-20 by ZHENG Zhong
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\chapter{Model of the Behavior of Stock Prices}

Any variable whose value changes over time in an uncertain way is said to follow a stochastic
process. Stochastic processes can be classified as discrete time or continuous time. A discrete-time
stochastic process is one where the value of the variable can change only at certain fixed points in
time, whereas a continuous-time stochastic process is one where changes can take place at any time.
Stochastic processes can also be classified as continuous variable or discrete variable. In a
continuous-variable process, the underlying variable can take any value within a certain range,
whereas in a discrete-variable process, only certain discrete values are possible.

This chapter develops a continuous-variable, continuous-time stochastic process for stock prices.

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\section{The Markov Property}

A Markov process is a particular type of stochastic process where only the present value of a
variable is relevant for predicting the future. The past history of the variable and the way that
the present has emerged from the past are irrevalent.

Stock prices are usually assumed to follow a Markov process. The Markov property of stock prices is
consistent with the weak form of market efficiency. This states that the present price of a stock
impounds all the information contained in a record of past prices.

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\section{Wiener Process}

A variable $z$ follows a Wiener process if it has the following two properties:

\emph{Property 1.} The change $\Delta z$ during a small period of time $\Delta t$ is:

\begin{equation}
  \Delta z = \epsilon \sqrt{\Delta t}
\end{equation}

where $\epsilon$ is a random drawing from a standardized normal distribution $\phi(0, 1)$.

\emph{Property 2.} The values of $\Delta z$ for any two different short intervals of time $\Delta t$
are independent.

It follows from property 1 that $\Delta t$ itself has a normal distribution with:

\begin{my_itemize}
  \item mean of $\Delta z$ = $0$
  \item standard deviation of $\Delta z$ = $\sqrt{\Delta t}$
  \item variance of $\Delta z$ = $\Delta t$
\end{my_itemize}

Property 2 implies that $z$ follows a Markov process.

Consider the increase in the value of $z$ during a relatively long period of time, $T$. This can be
denoted by $z(T) - z(0)$. It can be regarded as the sum of the increases in $z$ in $N$ small time
intervals of length $\Delta t$, where $N = T / \Delta t$. Thus:

\begin{equation}
  z(T) - z(0) = \sum_{i=1}^N \epsilon_i \sqrt{\Delta t}
\end{equation}

where the $\epsilon_i$ ($i = 1, 2, ..., N$) are random drawings from $\phi(0, 1)$. From property 2
of Wiener processes, the $\epsilon_i$'s are independent of each other. When we add two independent
normal distributions, the result is a normal distribution in which the mean is the sum of the means
and the variance is the sum of the variances. It follows that $z(T) - z(0)$ is normally distributed
with:

\begin{my_itemize}
  \item mean of $z(T) - z(0)$ = $0$
  \item variance of $z(T) - z(0)$ = $N \Delta t$ = $T$
  \item standard deviation of $z(T) - z(0)$ = $\sqrt{T}$
\end{my_itemize}

A Wiener process is the limit as $\Delta t \to 0$ of the process described above for $z$. Note that
the path followed by $z$ as the limit $\Delta t \to 0$ is approached is quite "jagged". This is
because the size of a movement in $z$ in time $\Delta t$ is proportional to $\sqrt{\Delta t}$ and,
when $\Delta t$ is small, $\sqrt{\Delta t}$ is much bigger than $\Delta t$.

Two intriguing properties of Wiener processes, related to this $\sqrt{\Delta t}$ property, are:

\begin{my_itemize}
  \item The expected length of the path followed by $z$ in any time interval is infinite.
  \item The expected number of times $z$ equals any particular value in any time interval is
        infinite.
\end{my_itemize}

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\section{Generalized Wiener Process}

The basic Wiener process, $dz$, that has been developed so far has a drift rate (i.e. average drift
per unit of time) of zero and a variance rate (e.g. variance per unit of time) of $1.0$. The drift
rate of zero means that the expected value of $z$ at any future time is equal to its current value.
The variance rate of $1.0$ means that the variance of the change in $z$ in a time interval of length
$T$ equals $T$.

A \emph{generalized Wiener process} for a variable $x$ can be defined in terms of $dz$ as follows:

\begin{equation}
  dx = a~dt + b~dz
  \label{generalized_wiener_process}
\end{equation}

where $a$ and $b$ are constants.

To understand equation \eqref{generalized_wiener_process}, it is useful to consider the two
components on the right-hand side separately. The $a~dt$ term implies that $x$ has an expected drift
rate of $a$ per unit of time. Without the $b~dz$ term, the equation is:

\[ dx = a~dt \]

which implies that:

\[ \frac{dx}{dt} = a \]

Integrating with respect to time, we get:

\[ x = x_0 + at \]

where $x_0$ is the value of $x$ at time zero. In a period of time of length $T$, the value of $x$
increases by an amount $aT$. The $b~dz$ term on the right-hand side can be regarded as adding noise
or variability to the path followed by $x$. The amount of this noise or variability is $b$ times a
Wiener process. A Wiener process has a standard deviation of $1.0$. It follows that $b$ times a
Wiener process has a standard deviation of $b$. In a small time interval $\Delta t$, the change
$\Delta x$ in the value of $x$ is:

\[ \Delta x = a~\Delta t + b~\epsilon~\sqrt{\Delta t} \]

where $\epsilon$ is a random drawing from a standardized normal distribution. Thus $\Delta x$ has a
normal distribution with:

\begin{my_itemize}
  \item mean of $\Delta x$ = $a \Delta t$
  \item standard deviation of $\Delta x$ = $b \sqrt{\Delta t}$
  \item variance of of $\Delta x$ = $b^2 \Delta t$
\end{my_itemize}

Similar arguments to those given for a Wiener process show that the change in the value of $x$ in
any time interval $T$, $x(T) - x(0)$, is normally distributed with:

\begin{my_itemize}
  \item mean of $x(T) - x(0)$ = $aT$
  \item standard deviation of $x(T) - x(0)$ = $b \sqrt{T}$
  \item variance of of $x(T) - x(0)$ = $b^2 T$
\end{my_itemize}

Thus, the generalized Wiener process given in equation \eqref{generalized_wiener_process} has an
expected drift rate of $a$ and a variance rate of $b^2$.

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\section{It\^o Process and It\^o's Lemma}

An \emph{It\^o process} is a generalized Wiener process in which the parameters $a$ and $b$ are
functions of the value of the underlying variable $x$ and time $t$. Algebraically, an It\^o process
can be written:

\begin{equation}
  dx = a(x, t) dt + b(x, t) dz
  \label{ito_process}
\end{equation}

Both the expected drift rate and variance rate of an It\^o process are liable to change over time.
In a small time interval between $t$ and $t + \Delta t$, the variable changes from $x$ to
$x + \Delta x$, where:

\[ \Delta x = a(x, t) \Delta t + b(x, t) \epsilon \sqrt{\Delta t} \]

This relationship involves a small approximation. It assumes that the drift and variance rate of $x$
remain constant, equal to $a(x, t)$ and $b(x, t)^2$, respectively, during the time interval between
$t$ and $t + \Delta t$.

Suppose that the value of a variable $x$ follows the It\^o process. The variable $x$ has a drift
rate of $a$ and a variance rate of $b^2$. Thus, a function $G$ of $x$ and $t$ follows the process:

\begin{equation}
  dG = \Big(   \frac{\partial G}{\partial x} a
             + \frac{\partial G}{\partial t}
             + \frac{1}{2} \frac{\partial^2 G}{\partial x^2} b^2
       \Big)~dt
     + \frac{\partial G}{\partial x}~b~dz
  \label{ito_lemma}
\end{equation}

where the $dz$ is the same Wiener process as in equation \eqref{ito_process}. Thus, $G$ also follows
an It\^o process. It has a drift rate of:

\[
    \frac{\partial G}{\partial x} a
  + \frac{\partial G}{\partial t}
  + \frac{1}{2} \frac{\partial^2 G}{\partial x^2} b^2
\]

and a variance rate of:

\[ \Big( \frac{\partial G}{\partial x} \Big)^2 b^2 \]

This is known as \emph{It\^o's lemma}.

Note that both $x$ and $G$ are affected by the same underlying source of uncertainty, $dz$.

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\section{The Process for Stock Prices}

A key aspect of stock prices is that, the expected percentage return required by investors from a
stock is independent of the stock price. So we assume that the expected return (i.e. expected
drift divided by the stock price) is constant. If $S$ is the stock price at time $t$, the expected
drift rate in $S$ should be assumed to be $\mu S$ for some constant parameter $\mu$. This means that
in a short interval of time, $\Delta t$, the expected increase in $S$ is $\mu S \Delta t$. The
parameter $\mu$ is the expected rate of return on the stock, expressed in decimal form.

If the volatility of the stock price is always zero, this model implies that:

\[ \Delta S = \mu S \Delta t \]

In the limit as $\Delta t \to 0$:

\[ dS = \mu S dt \]

or:

\[ \frac{dS}{S} = \mu dt \]

Integrating between time zero and time $T$, we get:

\[ S_T = S_0 e^{\mu T} \]

where $S_0$ and $S_T$ are the stock price at time zero and time $T$. This equation shows that, when
the variance rate is zero, the stock price grows at a continuously compounded rate of $\mu$ per unit
of time.

In practice, a reasonable assumption about the volatility of stock price is that, the variability of
the percentage return in a short period of time, $\Delta t$, is the same regardless of the stock
price. This suggests that the standard deviation of the change in a short period of time $\Delta t$
should be proportional to the stock price and leads to the model:

\begin{equation}
  dS = \mu S dt + \sigma S dz
  \label{stock_price_behavior_model_continuous}
\end{equation}

or:

\begin{equation}
  \frac{dS}{S} = \mu dt + \sigma dz
  \label{stock_price_behavior_model_continuous_2}
\end{equation}

Equation \eqref{stock_price_behavior_model_continuous_2} is the most widely used model of the stock
price behavior. It is known as \emph{geometric Brownian motion}. The variable $\sigma$ is the
volatility of the stock price. The variable $\mu$ is its expected rate of return.

Note: by comparing equation \eqref{stock_price_behavior_model_continuous} with the It\^o process
described in equation \eqref{ito_process}, we can see that $S$ follows a special It\^o process where
the instantaneous drift rate is $\mu S$, and the instantaneous variance rate is $\sigma^2 S^2$.

The discrete version of the model is:

\begin{equation}
  \Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
  \label{stock_price_behavior_model_discrete}
\end{equation}

or:

\begin{equation}
  \frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}
  \label{stock_price_behavior_model_discrete_2}
\end{equation}

The variable $\Delta S$ is the change in the stock price $S$ in a small time interval $\Delta t$,
and $\epsilon$ is a random drawing from a standardized normal distribution $\phi(0, 1)$. The
parameter $\mu$ is the expected rate of return per unit of time from the stock, and the parameter
$\sigma$ is the volatility of the stock price. Both of these parameters are assumed constant.

The left-hand side of equation \eqref{stock_price_behavior_model_discrete_2} is the return provided
by the stock in a short period of time $\Delta t$. The term $\mu \Delta t$ is the expected value of
this return, and the term $\sigma \epsilon \sqrt{\Delta t}$ is the stochastic component of the
return. The variance of the stochastic component (and therefore of the whole return) is
$\sigma^2 \Delta t$. This is consistent with the definition of the volatility $\sigma$: $\sigma$ is
such that $\sigma \sqrt{\Delta t}$ is the standard deviation of the return in a short time period
$\Delta t$.

Equation \eqref{stock_price_behavior_model_discrete_2} shows that $\Delta S / S$ is normally
distributed with mean $\mu \Delta t$ and a standard deviation $\sigma \sqrt{\Delta t}$. In other
words:

\begin{equation}
  \frac{\Delta S}{S} \sim \phi (\mu \Delta t, \sigma \sqrt{\Delta t})
\end{equation}

The process for stock prices involves two parameters: $\mu$ and $\sigma$. The parameter $\mu$ is the
expected continuously compounded return earned by an investor per year. Most investors require
higher expected returns to induce them to take higher risks. It follows that the value of $\mu$
should depend on the risk of the return from the stock. It should also depend on the level of
interest rates in the economy. The higher the level of interest rates, the higher the expected
return required on any given stock.

Fortunately, we do not have to concern ourselves with the determinants of $\mu$ in any detail,
because the value of a derivative dependent on a stock is, in general, independent of $\mu$. The
parameter $\sigma$, the stock price volatility, is, by contrast, critically important to the
determination of the value of most derivatives. Typical values of $\sigma$ for a stock are in the
range $0.20$ to $0.50$ (i.e. $20\%$ to $50\%$).

The standard deviation of the proportional change in the stock price in a small interval of time
$\Delta t$ is $\sigma \sqrt{\Delta t}$. As a rough approximation, the standard deviation of the
proportional change in the stock price over a relatively long period of time $T$ is
$\sigma \sqrt{T}$. This means that, as an approximation, volatility can be interpreted as the
standard deviation of the change in the stock price in one year.

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\section{The Lognormal Property}

We now use It\^o's lemma to derive the process followed by $ln S$ when $S$ follows the process in
equation \eqref{stock_price_behavior_model_continuous}. Define:

\[ G = ln S \]

From It\^o's lemma, we have:

\[
  dG = \Big( \frac{\partial G}{\partial S} \mu S
           + \frac{\partial G}{\partial t}
           + \frac{1}{2} \frac{\partial^2 G}{\partial S^2} \sigma^2 S^2
       \Big) dt
     + \frac{\partial G}{\partial S} \sigma S dz
\]

Because:

\[
  \frac{\partial G}{\partial S} = \frac{1}{S},
  \frac{\partial^2 G}{\partial S^2} = -\frac{1}{S^2},
  \frac{\partial G}{\partial t} = 0
\]

Thus the process followed by $G$ is:

\begin{equation}
  dG = \Big( \mu - \frac{\sigma^2}{2} \Big) dt + \sigma dz
\end{equation}

Because $\mu$ and $\sigma$ are constant, this equation indicates that $G = ln S$ follows a
generalized Wiener process. It has constant drift rate $\mu - \sigma^2 / 2$ and constant variance
rate $\sigma^2$. The change in $ln S$ between time zero and some future time $T$ is therefore
normally distributed with mean:

\[ \Big( \mu - \frac{\sigma^2}{2} \Big) \]

and variance:

\[ \sigma^2 T \]

This means that:

\begin{equation}
  ln S_T - ln S_0 \sim \phi \Big( \big( \mu - \frac{\sigma^2}{2} \big) T, \sigma \sqrt{T} \Big)
  \label{distr_ln_ST_ln_S0}
\end{equation}

or:

\begin{equation}
  ln S_T \sim \phi \Big( ln S_0 + \big( \mu - \frac{\sigma^2}{2} \big) T, \sigma \sqrt{T} \Big)
  \label{distr_ln_ST}
\end{equation}

where $S_T$ is the stock price at a future time $T$, $S_0$ is the stock price at time zero, and
$\phi(m, s)$ denotes a normal distribution with mean $m$ and standard deviation $s$.

Equation \eqref{distr_ln_ST} shows that $ln S_T$ is normally distributed. A variable has a lognormal
distribution if the natural logarithm of the variable is normally distributed. The model of the
stock price behavior therefore implies that a stock price at time $T$, given its price today, is
lognormally distributed. The standard deviation of the logarithm of the stock price is
$\sigma \sqrt{T}$. It is proportional to the square root of how far ahead we are looking.

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